P Values Robin Beaumont 8/2/2012 With much help from Professor Chris Wilds material University of Auckland

Where do they fit in!

Putting it all together

P Value A P value is a special type of probability: It considers more than one outcome (one event can have more than one outcome) Is a conditional probability A typical probability value: 0.25 A probability must be between 0 and 1 Probability Values e.g. Probability of winning the lottery yes no All possible outcomes at any one time must add up to 1

Probabilities are rel. frequencies

Multiple outcomes at any one time Probability Density Function Scores Probability The total area = 1 total 48 scores Density p(score<45) = area A A p(score > 50) = area B B P(score 50) = Just add up the individual outcomes

The probability of a value more extreme? The ‘more extreme’ idea Normal Distribution: x Chi-Squared Distribution: df = 9 Density

= Conditional Probability P(male) P(Male AND bearded) = 0.6 x = 0.2 What happens if events affect each other? Multiple each branch of the tree to get end value P(bearded|male) P(bearded AND male) = P(male) x P(bearded| male) Example from Taylor – From patient data to medical knowledge p in a room : 8 female + 12 male 4 of which have a beard P(bearded) = 4/20 = 0.2 P(male) = 12/20 =.6 So does the probability of being a bearded male = 0.2 x 0.6 = 0.12 NO 20 12/20 =.6 P(female) 8/20 = /12 =.3333 P(clear|male)

Screening Example 0.1% of the population (i.e 1 in a thousand) carry a particular faulty gene. A test exists for detecting whether an individual is a carrier of the gene. In people who actually carry the gene, the test provides a positive result with probability % of the time we get the correct result In people who don’t carry the gene, the test provides a positive result with probability % of the time we get a incorrect positive result Let G = person carries gene P = test is positive for gene N = test is negative for gene Errors Given that someone has a positive result, find the probability that they actually are a carrier of the gene. We want to find Need P(P) looking at the two P(P) branches P(P) = P(G and P) + P(G' and P) = = P( P | G) P(P | G) ≠ P (G | p) ORDER MATTERS

= Conditional Probability P(disease) Disease X AND test+ Disease / Test P(test+|disease)

The probability of obtaining the hypothesised value GIVEN THAT we obtained the summary value x Summary value=x observed | hypothesised Hypothesised value The probability of obtaining summary value x GIVEN THAT I have this hypothesised value Hypothesised value summary value=x X P(summary value=x|hypothesised value) P(hypothesised value|summary value=x)

Combining conditional probability + multiple outcomes = P value A P value is a conditional probability considering a range of outcomes Chi-Squared Distribution: df = 9 Density Here we have a probability distribution of possible observed values for the chi- square summary statistic GIVEN THAT The hypothesised value is ZERO P value = P(observed chi square value or one more extreme |value = 0) The blue bit presents all those values greater than 15 Area = This is the P value

Probability summary All outcomes at any one time add up to 1 Probability histogram = area under curve =1 -> specific areas = sets of outcomes “More extreme than x” Conditional probability –– ORDER MATTERS A P value is a conditional probability which considers a range of outcomes

Putting it all together

Populations and samples Ever constant at least for your study! = Parameter estimate = statistic

One sample

Size matters – single samples

Size matters – multiple samples

We only have a rippled mirror

Standard deviation - individual level = measure of variability 'Standard Normal distribution' Total Area = = SD value 68% 95% 2 Area: Between + and - three standard deviations from the mean = 99.7% of area Therefore only 0.3% of area(scores) are more than 3 standard deviations ('units') away. - But does not take into account small sample size = t distribution Defined by sample size aspect ~ df Area! Wait and see

Sampling level -‘accuracy’ of estimate From: = 5/√5 = SEM = 5/√25 = 1 We can predict the accuracy of your estimate (mean) by just using the SEM formula. From a single sample Talking about means here

Example - Bradford Hill, (Bradford Hill, 1950 p.92) mean systolic blood pressure for 566 males around Glasgow = mm. Standard deviation =13.05 Determine the ‘precision’ of this mean. “We may conclude that our observed mean may differ from the true mean by as much as ± (.5485 x 4) but not more than that in around 95% of observations. page 93. [edited]

Sampling summary The SEM formula allows us to: predict the accuracy of your estimate ( i.e. the mean value of our sample) From a single sample Assumes Random sample

Variation what have we ignored! Onto Probability now

Putting it all together

Statistics Summary measure – SEM, Average etc T statistic – different types, simplest: So when t = 0 means 0/anything = estimated and hypothesised population mean are equal So when t = 1 observed different same as SEM So when t = 10 observed different much greater than SEM

T statistic example Serum amylase values from a random sample of 15 apparently healthy subjects. The mean = 96 SD= 35 units/100 ml. How likely would such a sample be obtained from a population of serum amylase determinations with a mean of 120. (taken from Daniel 1991 p.202 adapted) This looks like a rare occurrence? But for what A population value = the null hypothesis

t density:s x = n = t Shaded area = Original units: 0 Serum amylase values from a random sample of 15 apparently healthy subjects. mean =96 SD= 35 units/100 ml. How likely would such a sample be obtained from a population of serum amylase determinations with a mean of 120. (taken from Daniel 1991 p.202 adapted) What does the shaded area mean! Given that the sample was obtained from a population with a mean of 120 a sample with a T (n=15) statistic of or or one more extreme will occur 1.8% of the time = just under two samples per hundred on average... Given that the sample was obtained from a population with a mean of 120 a sample of 15 producing a mean of 96 (120-x where x=24) or 144 (120+x where x=24) or one more extreme will occur 1.8% of the time, that is just under two samples per hundred on average. But it this not a P value p = 2 · P(t (n−1) < t| H o is true) = 2 · [area to the left of t under a t distribution with df = n − 1]

P value and probability for t statistic p value = 2 x P(t (n-1 ) values more extreme than t (n-1 ) | H o is true ) = 2 · [area to the left of t under a t distribution with n − 1 shape] A p value is a special type of probability with: Multiple outcomes + conditional upon the specified parameter value

Putting it all together Do we need it!

Rules t density:s x = n = t Shaded area = Original units: 0 Set a level of acceptability = critical value (CV)! Say one in twenty 1/20 = Or 1/100 Or 1/1000 or.... If our result has a P value of less than our level of acceptability. Reject the parameter value. Say 1 in 20 (i.e.CV=0.5) Given that the sample was obtained from a population with a mean (parameter value) of 120 a sample with a T (n=15) statistic of or or one more extreme with occur 1.8% of the time, This is less than one in twenty therefore we dismiss the possibility that our sample came from a population mean of What do we replace it with?

Fisher – only know and only consider the model we have i.e. The parameter we have used in our model – when we reject it we accept that any value but that one can replace it. Neyman and Pearson + Gossling Must have an alternative specified value for the parameter

If there is an alternative - what is it – another distribution! Power – sample size Affect size – indication of clinical importance: Serum amylase values from a random sample of 15 apparently healthy subjects. mean =96 SD= 35 units/100 ml. How likely would such a sample be obtained from a population of serum amylase determinations with a mean of 120. (taken from Daniel 1991 p.202 adapted)

α = the reject region = 120 = 96 Correct decisions incorrect decisions

Insufficient power – never get a significant result even when effect size large Too much power get significant result with trivial effect size

Life after P values Confidence intervals Effect size Description / analysis Bayesian statistics - qualitative approach by the back door! Planning to do statistics for your dissertation? see: My medical statistics courses: Course 1: YouTube videos to accompany course 1: Course 2: YouTube videos to accompany course 2:

Your attitude to your data

Where do they fit in!